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mathematics
cambridge international as & a level mathematics probability & statistics
Cambridge International AS And A Level Mathematics Pure Mathematics 1 Coursebook 1st Edition Sue Pemberton, Julian Gilbey - Solutions
Find the value of a and the value of b.f(x) = x3 + ax2 + bx − 5f(x) has a factor of x − 1 and leaves a remainder of −6 when divided by x + 1.
Solve.a. |2x + 1|< 3b. |2 − x| < 4c. |3x − 2|>7
Solve the inequality |2x − 3| < |2 − x|.
Describe fully the transformation (or combination of transformations) that maps the graph of y = |x| onto each of these functions.a. y = |x + 1| + 2 b. y = |x − 5| − 2c. y = 2 − |x|d. y = |2x| − 3e. y = 1 − |x + 2|f. y = 5 − 2|x|
a. Use algebraic division to show that x − 2 is a factor of 2x3 + 7x2 − 43x + 42.b. Hence, factorise 2x3 + 7x2 − 43x + 42 completely.
Given that x − 3 is a factor of x3 + ax2 + bx − 30, express a in terms of b.
The polynomial 3x3 + ax2 + bx + 8, where a and b are constants, is denoted by f(x). It is given that x + 2 is a factor of f(x), and that when f(x) is divided by x − 1 the remainder is 15. Find the value of a and the value of b.
Solve the equation |x2 − 14| = 11.
Sketch the graphs of each of the functions in question 3. For each graph, state the coordinates of the vertex.
Solve. a. |2x − 5| < x − 2 b. |3 + x| > 4 − 2x c. |x − 7| − 2x < 4
Solve the simultaneous equations.a. x + 2y = 8|x+ 2|+ y = 6b. 3x + y = 0y = |2x2 - 5|
a. Use algebraic division to show that 2x + 1 is a factor of 12x3 + 16x2 − 3x − 4.b. Hence, solve the equation 12x3 + 16x2 − 3x − 4 = 0
The polynomial ax3 − 13x2 − 41x − 2a, where a is a constant, is denoted by p(x).a. Given that (x − 4) is a factor of p(x), find the value of a.b. When a has this value, factorise p(x) completely.
Find the range of function f. Find the range of solution f.f(x) = |5 − 2x| + 3 for 2 < x < 8
Solve.a. |x + 4| < |2x| b. |2x − 5| > 3 − x c. |3x − 2| < 1 − 3xd. |x − 1| > 2 x − 4 e. 3|3 − x| > |2x − 1| f. |3x − 5| < 2|2 − x|
Solve the equation 5|x - 1|2 + 9|x - 1|-2 = 0
a. Use algebraic division to show that x + 1 is a factor of x3 − x2 + 2x + 4.b. Hence, show that there is only one real root for the equation x3 − x2 + 2x + 4 = 0.
The general formula for solving the quadratic equation ax2 + bx + c = 0 isThere is a general formula for solving a cubic equation. Find out more about this formula and use it to solve 2x3 − 5x2 − 28x + 15 = 0. X = -b ± √b² - 4ac 2a
Given that x − 3 and 2x + 1 are factors of 2x3 + px2 + (2q − 1)x + q:a. Find the value of p and the value of q.b. Explain why x + 2 is also a factor of the expression.
The polynomial 6x3 − 23x2 − 38x + 15 is denoted by f(x).a. Show that (x − 5) is a factor of f(x) and hence factorise f(x) completely.b. Write down the roots of f (|x|) = 0.
a. Sketch the graph of y = 2 |x − 2| + 1 for −2 < x < 6, showing the coordinates of the vertex and the y-intercept.b. On the same diagram, sketch the graph of y = x + 2.c. Use your graph to solve the equation 2 |x − 2| + 1 = x + 2.
Solve |2x + 1|+ |2x − 1|> 5.
a. Solve the equation x2 − 5 x + 6 = 0.b. Use graphing software to draw the graph of y = x2 − 5 x + 6.c. Name the equation of the line of symmetry of the curve.
It is given that x + a is a factor of x3 + 4x2 + 7ax + 4a. Find the possible values of a.
The polynomial x3 + ax2 + bx + 2, where a and b are constants, is denoted by f(x). It is given that x − 2 is a factor of f(x), and that when f(x) is divided by x + 1 the remainder is 21.a. Find the value of a and the value of b.b. Solve the equation f(x) = 0, giving the roots in exact form.
The polynomial x3 − 5x2 + ax + b is denoted by f(x). It is given that (x + 2) is a factor of f(x) and that when f(x) is divided by (x − 1) the remainder is −6. Find the value of a and the value of b.
a. Sketch the graph of y =|x − 2| for −3 < x < 6, showing the coordinates of the vertex and the y-intercept.b. On the same diagram, sketch the graph of y = 1 − 2x.c. Use your graph to solve the equation |x − 2| = |1 − 2x|.
Solve the equation |2x + 1| + |2x − 1| = 3.
a. It is given that x + 1 is a common factor of x3 + px + q and x3 + (1 − p)x2 + 19x − 2q. Find the value of p and the value of a. b. When p and q have these values, factorise x3 + px + q and x3 + (1 − p)x2 + 19x − 2q completely.
The polynomial 2x3 + ax2 + bx + c is denoted by f(x). The roots of f(x) = 0 are −1, 2 and k. When f(x) is divided by x − 1, the remainder is 6.a. Find the value of k.b. Find the remainder when f(x) is divided by x + 2.
The polynomial x3 − 5x2 + 7x − 3 is denoted by p(x).a. Find the quotient and remainder when p(x) is divided by (x2 − 2x − 1).b. Use the factor theorem to show that (x − 3) is a factor of p(x).
a. Sketch the graph of y = |x + 1| + |x − 1|.b. Use your graph to solve the equation |x + 1| + |x − 1|= 4.
Solve the equation |3x − 2y − 11|+ 2√31 - 8x + 5y = 0.
Given that x − 1 and x + 2 are factors of x4 − x3 + px2 − 11x + q:a. Find the value of p and the value of q.b. When p and q have these values, factorise x4 − x3 + px2 − 11x + q completely.
The polynomial 4x4 + 4x3 − 7x2 − 4x + 8 is denoted by p(x).a. Find the quotient and remainder when p(x) is divided by (x2 − 1).b. Hence solve the equation 4x4 + 4x3 − 7x2 − 4x + 3 = 0.
Find the remainder when P(x) is divided by (x − 1).P(x) = 2x + 4x2 + 6x3 +…+ 100x50
Solve.a. x3 − 5x2 − 4x + 20 = 0 b. x3 + 5x2 − 17x − 21 = 0c. 2x3 − 5x2 − 13x + 30 = 0 d. 3x3 + 17x2 + 18x − 8 = 0e. x4 + 2x3 − 7x2 − 8x + 12 = 0f. 2x4 − 11x3 + 12x2 + x − 4 = 0
The polynomial x4 − 48x2 − 21x − 2 is denoted by f(x).a. Find the value of the constant k for which f(x) = (x2 + kx + 2)(x2 − kx − 1).b. Hence solve the equation f(x) = 0. Give your answers in exact form.
Find the value of a, the value of b and the value of c.P(x) = 3(x + 1)(x + 2)(x + 3) + a(x + 1)(x + 2) + b(x + 1) + cIt is given that when P(x) is divided by each of x + 1, x + 2 and x + 3 the remainders are −2, 2 and 10, respectively.
You are given that the equation x3 + ax2 + bx + c = 0 has three real roots and that these roots are consecutive terms in an arithmetic progression. Show that 2a3 + 27c = 9ab.
The polynomial 2x4 + 3x3 − 12x2 − 7x + a is denoted by p(x).a. Given that (2x − 1) is a factor of p(x), find the value of a.b. When a has this value, verify that (x + 3) is also a factor of p(x) and hence factorise p(x) completely.
Find the set of values for k for which the equation 3x4 + 4x3 − 12x2 + k = 0 has four real roots.
The polynomial 3x3 + ax2 − 36x + 20 is denoted by p(x).a. Given that (x − 2) is a factor of p(x), find the value of a.b. When a has this value, solve the equation p(x) = 0.
The polynomial 2x3 + 5x2 − 7x + 11 is denoted by f(x).a. Find the remainder when f(x) is divided by (x − 2).b. Find the quotient and remainder when f(x) is divided by (x2 − 4x + 2).
The polynomial ax3 + bx2 − x + 12 is denoted by p(x).a. Given that (x − 3) and (x + 1) are factors of p(x), find the value of a and the value of b.b. When a and b take these values, find the other linear factor of p(x).
The polynomial 2x3 + ax2 + bx + 6 is denoted by p(x).a. Given that (x + 2) and (x − 3) are factors of p(x), find the value of a and the value of b.b. When a and b take these values, factorise p(x) completely.
The polynomial 4x3 + kx2 − 65x + 18 is denoted by f(x).a. Given that (x + 2) is a factor of f(x), find the value of k.b. When k has this value, solve the equation f(x) = 0.c. Write down the roots of f(x2) = 0.
Without using a calculator, find the value of:a. log10 100b. log10 0.0001c. log10 (10√10)d. log10 (3√10)e. log (100 3√10)f. 100 logio 1000 10
The polynomial 2x3 − 9x2 + ax + b, where a and b are constants, is denoted by f(x). It is given that (x + 2) is a factor of f(x), and that when f(x) is divided by (x + 1) the remainder is 30.a. Find the value of a and the value of b.b. When a and b have these values, solve the equation f(x) = 0.
The polynomial x3 + 3x2 + 4x + 2 is denoted by f(x).i. Find the quotient and remainder when f(x) is divided by x2 + x − 1.ii. Use the factor theorem to show that (x + 1) is a factor of f(x).
The polynomial 4x3 + ax2 + 9x + 9, where a is a constant, is denoted by p(x). It is given that when p(x) is divided by (2x − 1) the remainder is 10.i. Find the value of a and hence verify that (x − 3) is a factor of p(x).ii. When a has this value, solve the equation p(x) = 0.
Solve each equation for the given domains.a. cosec (x − 30°) = 2 for 0° < x < 360°b. sec(2x + 60°) = −1.5 for 0° < x < 180°c.d. 2 cosec (2x − 1) = 3 for − π < x < π cot(x + 1) 4 = 2 for 0≤x≤2
Find the exact values of:a. sec 60°b. cosec 45°c. cot120°d. sec 300°e. cosec135°f. cot 330°g. sec150°h. cot(−30°)
Convert from logarithmic form to exponential form.a. log1010 000 = 4 b. log10x = 1.2 c. log10 = - 0.6
a. Sketch each of the following functions for the interval 0 < x < 2π.i. y = 1+ secxii. y = cot2xiii.iv. y = 1 − sec xv. y = 1+ cosec 1/2 xvi.b. Write down the equation of each of the asymptotes for your graph for part a vi. y = 2 cosec x TU 2
Solve each equation for 0 < x < 2π.a. cosec x = 2 b. sec x = −1c. cot x = 2d. 2 cot x + 5 = 0
Solve each equation for 0° < x < 180°.a. cosec 2x = 1.2 b. sec 2x = 4 c. cot 2x = 1 d. 2 sec 2x = 3
Prove each of these identities.a.b.c.d.e.f.f.e. 1 tanx + cot sin x cos x
Given that the function f is defined as f : x -> 10x − 3 for x € ℝ, find an expression for f −1(x).
Given that p and q are positive and that 4(log10 P)2 + 2(log10q)2 = 9, find the greatest possible value of p.
Solve each equation for −180° < x < 180°.a. cosec2 x = 4 b. sec2 x = 9 c. 9 cot2 1/2 x = 4d. sec x = cos x e. cosec x = sec x f. 2 tan x = 3 cosec x
Solve each equation for 0° < θ < 360°.a. 2 tan2 − 1 = secθb. 3 cosec2 = 13 − cotθc. 2 cot2 − cosec = 13d. cosecθ + cotθ = 2 sinθ e. tan2θ + 3secθ = 0 f. √3 sec2θ = cosecθ
Solve each equation for 0° < 180°.a. secθ = 3cosθ − tanθ b. 2 sec22θ = 3tan2θ + 1c. sec4θ + 2 = 6tan2θd. 2 cot2 2θ + 7 cosec 2θ = 2
Solve each equation for 0 < θ < 2π.a. tan2θ + 3secθ + 3 = 0b. 3cot2 θ + 5 cosecθ + 1= 0
Prove each of these identities.a. sin x + cos x cot x ≡ cosec x.b. cosec x − sin x ≡ cos x cot x.c. sec x cosec x − cot x ≡ tan x.d. (1+ secx)(cosec x − cot x) ≡ tan x.
Solve each equation for 0° < θ < 180°.a. 6sec3θ − 5sec2 θ − 8 secθ + 3 = 0b. 2 cot3 + 3cosec2 − 8 cot θ = 0
The line x + y = 4 meets the curveat the points A and B. Find the coordinates of the midpoint of AB. 5 y = 8 - - X
Solve the simultaneous equations.a. b.c.d.e.f.g.h.i.j.k.l.m.n.o. y = 6-x y = x²
Solve:a. x(x − 3) ≤ 0 b. (x − 3)(x + 2) > 0 c. (x − 6)(x − 4) ≤ 0d. (2x + 3)(x − 2) < 0 e. (5 − x)(x + 6) ≥ 0 f. (1 − 3x)(2x + 1) < 0
Find the values of k for which the line y = kx + 1 is a tangent to the curve y = x2 − 7x + 2.
Find the real values of x that satisfy the following equations.a. x4 − 13x2 + 36 = 0 b. x6 − 7x3 − 8 = 0 c. x4 − 6x2 + 5 = 0d. 2x4 − 11x2 + 5 = 0 e. 3x4 + x2 − 4 = 0 f. 8x6 − 9x3 + 1 = 0g. x4 + 2x2 − 15 = 0 h. x4 + 9x2 + 14 = 0 i. x8 − 15x4 − 16 =
A curve has equation y = 2xy + 5 and a line has equation 2x + 5y = 1. The curve and the line intersect at the points A and B. Find the coordinates of the midpoint of the line AB.
Solve:a.b.c.d.e.f. x- 6 x-5 = 0
Express each of the following in the form (x + a)2 + b.a. x2 − 6x b. x2 + 8x c. x2 − 3x d. x2 + 15xe. x2 + 4x + 8 f. x2 − 4x − 8g. x2 + 7x + 1 h. x2 − 3x + 4
Use the symmetry of each quadratic function to find the maximum or minimum points.Sketch each graph, showing all axes crossing points.a. y = x2 − 6x + 8 b. y = x2 = 5x − 14 c. y = 2x2 + 7x − 15 d. y = 12 + x − x2
Use the discriminant to determine the nature of the roots of 2 – 5x = 4 +*
Solve using the quadratic formula. Give your answer correct to 2 decimal places.a. x2 − 10x − 3 = 0 b. x2 + 6x + 4 = 0 c. x2 + 3x − 5 = 0d. 2x2 + 5x − 6 = 0 e. 4x2 + 7x + 2 = 0 f. 5x2 + 7x − 2 = 0
Solve:a. 2x − 9√x + 10 = 0 b. √x(√x + 1) = 6 c. 6x − 17√x + 5 = 0d. 10x + √x − 2 = 0 e. 8x + 5 = 14√xf. 35x+ 5 = 16
Solve by factorization. a. x2 + 3x − 10 = 0 b. x2 − 7x + 12 = 0c. x2 − 6x − 16 = 0d. 5x2 + 19x + 12 = 0 e. 20 − 7x = 6x2 f. x(10x − 13) = 3
The sum of two numbers is 26. The product of the two numbers is 153.a. What are the two numbers?b. If instead the product is 150 (and the sum is still 26), what would the two numbers now be?
Find the values of k for which the x-axis is a tangent to the curve y = x2 − (k + 3)x + (3k + 4).
Solve:a. x2 − 25 ≥ 0 b. x2 + 7x + 10 ≤ 0 c. x2 + 6x − 7 > 0d. 14x2 + 17x − 6 ≤ 0 e. 6x2 − 23x + 20 < 0 f. 4 − 7x − 2x2 < 0
Find the real roots of the equation 36 X + -4 = 25 2 X
a. Express 9x2 − 15x in the form (3x − a)2 − b.b. Find the set of values of x that satisfy the inequality 9x2 − 15x < 6.
Express each of the following in the form a(x + b)2 + c.a. 2x2 − 12x + 19 b. 3x2 − 12x − 1 c. 2x2 + 5x − 1 d. 2x2 + 7x + 5
Find the value of k for which the line x + ky = 12 is a tangent to the curveCan you explain graphically why there is only one such value of k? y = 5 x-2
a. Express 2x2 − 8x + 5 in the form a(x + b)2 + c, where a, b and c are integers.b. Write down the equation of the line of symmetry for the graph of y = 2x2 − 8x + 1.
A rectangle has sides of length x cm and (3x − 2) cm.The area of the rectangle is 63cm2.Find the value of x, correct to 3 significant figures.
The perimeter of a rectangle is 15.8cm and its area is 13.5cm2. Find the lengths of the sides of the rectangle.
Find the real solutions of the following equations.a. 8(x2 + 2x - 15) = 1b. 4(2x2 − 11 + 15) = 1c. 2(x2 − 4 + 6) = 8d.e. (x2 + 2x − 14)5 = 1f. (x2 - 7x + 11)8 = 1 3(2x² +9x+2) 9
Solve:a. x2 < 36 − 5x b. 15x < x2 + 56 c. x(x + 10) ≤ 12 − xd. x2 + 4x ,3(x + 2) e. (x + 3)(1 − x) < x − 1 f. (4x + 3)(3x − 1) < 2x(x + 3)g. (x + 4)2 ≥ 25 h. (x − 2)2 >14 − x i. 6x(x + 1) < 5(7 − x)
Express each of the following in the form a − (x + b)2.a. 4x − x2 b. 8x − x2 c. 4 − 3x − x2 d. 9 + 5x − x2
The equation x2 + bx + c = 0 has roots −5 and 7. Find the value of b and the value of c.
The curve y = 2 √x and the line 3y = x + 8 intersect at the points A and B.a. Write down an equation satisfied by the x-coordinates of A and B.b. Solve your equation in part a and, hence, find the coordinates of A and B.c. Find the length of the line AB.
Find the value of x, correct to 3 significant figures.Rectangle A has sides of length x cm and (2x − 4) cm.Rectangle B has sides of length (x + 1) cm and (5 − x) cm.Rectangle A and rectangle B have the same area.
Find the range of values of x for which 5 2x²+x-15
a. Express 2x2 + 9x + 4 in the form a(x + b)2 + c, where a, b and c are constants.b. Write down the coordinates of the vertex of the curve y = 2x2 +9x +4, and state whether this is a maximum or a minimum point.
The sum of the perimeters of two squares is 50 cm and the sum of the areas is 93.25cm2. Find the side length of each square.
The graph shows y = ax + b √x + c for x ≥ 0. The graph crosses the x-axis at the points (1, 0 ) and (49/4, 0) and it meets the y-axis at the point (0, 7). Find the value of a, the value of b and the value of c. 7 49 4 我
The line y = k − 3x is a tangent to the curve x2 + 2xy − 20 = 0.a. Find the possible values of k.b. For each of these values of k, find the coordinates of the point of contact of the tangent with the curve.
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